Hierarchical Risk Parity
Summary
I compare the hierarchical risk parity (HRP) portfolio to six other portfolio strategies: inverse volatility (IV), equal weight (EW), global minimum variance (GMV), maximum diversification (MD) equal risk contribution (ERC) and maximum sharpe ratio (MSR). I report out-of sample performance, portfolio composition and correlation in two asset sets: S&P500 stocks from Dec 1991 thru Nov 2016 and in global futures from Jan 2002 thru Apr 2016. I simulate both the original version of the HRP portfolio using variance risk measure (HRP-VAR) as well as a similarly defined portfolio using the volatility risk measure (HRP-VOL).
HRP-VOL turns out to be very closely related the ERC portfolio. Performances are highly correlated, and both portfolios allocate funds in similar proportions to assets and reach a similar performance. Both are essentially mechanisms to allocate a share of volatility risk to every asset, leading to the same benefits (balanced allocation, smaller dependence on estimation error) and drawbacks (sub-optimal diversification). Furthermore, the benefits of the HRP portfolio claimed by the author - reduced sensitivity to parameter estimation leading to increased robustness in large asset sets - apply to both strategies and do not discriminate between them.
HRP-VAR, the original hierarchical risk parity portfolio based on variance risk, outperforms its declared benchmark, the GMV portfolio, in global futures. This is of limited importance, as the GMV portfolio performs poorly in this data set in the first place and since it concentrates the allocation in a small number of assets with lowest variance. The outperformance is caused by a a more balanced allocation of the HRP-VAR portfolio. I imagine that an ERC portfolio based on the variance risk measure would lead to a similar outcome.
While I sympathize with the algorithmic approach of the portfolio, I do not see a clear benefit over the already established and much better researched alternatives.
Allocation Strategies
Hierarchical Risk Parity (HRP)
López de Prado (2016) orders assets hierarchically and then recursively cuts asset in halves, allocating an equal share of risk to both halfs.
- Hierarchically cluster assets based on distance \(d_{i,j}\) \[ d_{i,j} = \sqrt{\frac{1}{2}(1-\rho_{i,j})}\quad\quad \rho_{i,j}\text{: correlation between assets } i \text{ and } j \]
- Order assets in sequence of the dendogramm, see figures 1-4 for illustration
- Allocate risk through recursive bi-section:
- split assets into halves
- assign equal variance \(\sigma^2 = \mathbf{w'\Sigma w}\) using \(w_i = 1 / \sigma_i^2\) to each half, and
- repeat above steps for each half of the assets.
He claims that this algorithm reduces sensitivity to paramter estimation, leads to more stable portfolios and is particularely well suited for a large number of portfolio constituents.
López de Prado (2016) uses the variance risk measure \(\sigma^2\) and uses the global minimum variance (GMV) portfolio as a benchmark. I name his definition HRP-VAR and introduce a similar portfolio, HRP-VOL, based on volatility instead of variance. Risk is now measured as \(\sigma = \sqrt{\mathbf{w'\Sigma w}}\) using \(w_i = 1 / \sigma_i\).
Dendrogram
Figures 1-4 show the dendrogram of hierarchical clusters computed with global futures at four moments throughout our study period. Clusters are based on the previous 3 years of returns. Colors assign futures instruments to asset classes equities, commodities and fixed income.
Observations:
- The hierarchical clustering nicely arranges asset of the same asset class side-by side.
- Assets and entire clusters can, however, switch positions from one iteration to the next.
- Splitting assets in two halves assigns assets arbitrarily to two halves, once time grouping equities and half the commodities together, another time grouping half the equities and fixed income together.
2002-01-31
Figure 1: Hierarchical clustering of Global Futures as of 2002-01-31 based on three years of daily returns.
2006-10-31
Figure 2: Hierarchical clustering of Global Futures as of 2006-10-31 based on three years of daily returns.
2011-07-29
Figure 3: Hierarchical clustering of Global Futures as of 2011-07-29 based on three years of daily returns.
2016-04-29
Figure 4: Hierarchical clustering of Global Futures as of 2016-04-29 based on three years of daily returns.
Other objectives
- The maximum decorrelation (MDC) portfolio minimizes portfolio variance under the assumption of identical volatility for all assets.
\[ \min_{w \geq 0} \mathbf{w}'\mathbf{P}\mathbf{w} \]
The inverse volatility (IV) portfolio scales assets to equal single-asset volatility. \[ w_i = \frac{1}{\sigma_r} \]
The maximum diversified (MD) portfolio proposed by Choueifaty and Coignard (2008) maximizes the diversification ratio defined as the ratio between weighted sum of single asset volatility over portfolio volatility. \[ \max_{\mathbf{w}\geq 0} \text{DR}, \quad \text{DR} = \frac{\textbf{w}'R(\textbf{r})}{R(\textbf{w}'\textbf{r})} = \frac{\mathbf{w}'\mathbf{\sigma}_r}{\sqrt{\mathbf{w}'\mathbf{\Sigma}\mathbf{w}}} \]
The equal risk contribution (ERC) portfolio formalized by Maillard, Roncalli, and Teïletche (2010) aligns the contribution of each asset to the total portfolio risk. \[ \text{RC}_j = \text{RC}_i = w_i\frac{\partial R(\mathbf{w}'\mathbf{r})}{\partial w_i} = \frac{1}{2} w_i (\mathbf{\Sigma}\mathbf{w})_i \]
The equal weight (EW) portfolio allocates an equal amount to all assets: \[ w_i = \frac{1}{N} \]
The Maximum sharpe ratio (MSR) portfolio maximizes the portfolio’s excess return divided by volatility: \[ \max_{w\geq 0} \frac{\mathbf{w}'\;\mathbb{E}\left[\mathbf{r} - r_f\right]}{\sqrt{\mathbf{w}'\Sigma\mathbf{w}}} \]
Scaling
Portfolios are calibrated subject to two alternative investment constraints:
- Fully invested \[ s.t.\;\;\mathbf{w}'\mathbf{1}=1 \]
- Constant 10% annualized target volatility \[ s.t.\;\;\sqrt{\mathbf{w}'\mathbf{\Sigma}\mathbf{w}} = \sigma_{\text{target}} \]
Results
Risk and Performance
Figures 5-8 and tables 3-5 report out-of-sample risk and performance of the portfolio strategies in both asset sets. Portfolios are calibrated monthly based on the last three years of underlying market returns. Stock returns are reported in excess of the 6-monht T-bill rate. Futures returns are considered to be excess returns. Execution costs are not taken into account.
Observations:
- The HRP-VAR portfolio outperforms its benchmark, the GMV portfolio, in the global futures data set. Performance in single stocks is very similar between the two.
- The HRP-VOL portfolio performs similarly as the ERC portfolio. This is not surprising since both effectively allocate equal risk contribution to assets.
Global Futures - 10% Target Vola
Figure 5: Rolling one-year out-of-sample return and volatility. Execution cost is not taken into account. The plot is interactive, lines can be removed/restored by selecting the legend.
Table 1: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Execution cost is not taken into account. MSE reports the mean squared difference to the 10% volatility target in above figure.